bangkok:
First, the numbering confusion. Since the beginning of recorded history, or at least the middle of the 19th century, S&W have stamped their serial numbers on the bottom of the grip frame. On some guns they also stamped it on the cylinder face and on the underside of the barrel. In the 60s or seventies ( I believe the seventies, but could be wrong) with the trend towards large target grips that covered the bottom of the frame, they also began stamping serial numbers on the frame under the crane where the model number is. They kept the serial number on the bottom of the grip frame. So over the last 150 odd years serial numbers have been duplicated in several places, but always there has been a serial number on the bottom of the grip frame. So that's where we look for the serial number if there is any doubt.
Other numbers also get stamped on the gun during the manufacturing process. Some manufactured during the two world wars have military inspector stamps on them. The most common stampings seen however are what are known as assembly numbers. These are numbers stamped on parts during the manufacturing process, I guess a combination of "Kilroy was here" & "These parts are fitted, do not separate". These numbers are known as assembly numbers.
An example: My 1973 manufactured Model 66 has serial number 4K19xxx on the bottom of the grip frame & under the crane. It also has the assembly number 11155 & a large S stamped on the crane proper. I've seen a pic of another frame that had several different assembly numbers stamped under the grips.
But once the gun is assembled only the serial number matters. That is the number S&W records when it ships the gun, that's the number the BATF look for.
Now getting back to your model 36. The early J frames were serial numbered from 1 to 786544 between 1950 & 1969. This covered the models 36, 37, 38, 49 & 50. So when you gave me a number in the 82xxx range I looked up the early J frame list in the Standard Catalog of S&W and came up with 1956.
It appears that the number you quoted was an assembly number. Your actual serial number is J638xxx. In 1969 S&W started the J frame serial number range over at J1 and continued on up from there.
Again referring to the Standard Catalog of S&W, J638xxx falls into the serial range for 1977-1978, being J610001to J670000.